## Good Physics Books For a Math PhD Student? 418

An anonymous reader writes

*"As a third-year PhD math student, I am currently taking Partial Differential Equations. I'm working hard to understand all the math being thrown at us in that class, and that is okay. The problem is, I have never taken any physics anywhere. Most of the problems in PDEs model some sort of physical situation. It would be nice to be able to have in the back of my mind where this is all coming from. We constantly hear about the heat equation, wave equation, gravitational potential, etc. I'm told I should not worry about what the equations describe and just learn how to work with them, but I would rather not follow that advice. Can anyone recommend physics books for someone in my position? I don't want to just pick up a book for undergrads. Perhaps there are things out there geared towards mathematicians?"*
## I'm gunna say this once.. (Score:5, Funny)

Get back to writing your thesis.

Slacker.

## Re:I'm gunna say this once.. (Score:4, Funny)

I'd rather not follow that advice.

## Re:I'm gunna say this once.. (Score:5, Funny)

That proves it! Only a PhD student would say that.

## Re: (Score:2)

In the UK (and Europe I believe), it's normal to be finishing your thesis at the end of the 3rd year/first half of 4th year, I took 4 days more than 3 years. But even as a chemistry student, I did partial diff equations in the 1st year of my undergraduate (ie aged 18). So what kind of university are you at?

## Re:I'm gunna say this once.. (Score:4, Interesting)

> So what kind of university are you at?

An American one. They specialize *much* later than the English [who are down to 3 subjects, eg Maths, More Maths, and Physics] for A levels [at 16]. Once at university Americans still have to take a relatively broad range of classes for the first 2 years. So, even in good US universities the first year or two only gets them up to A level standard. Top performers make up for it later on because after being held back from focusing only on what interests them for so long its such a relief to be able to concentrate that they really get stuck into it.

## Standards have slipped then... (Score:3, Informative)

I graduated in 1985 with a BS in Math & Chemistry. Partial Differential Equations was a required course back then, and the school I attended was nothing special in terms of what they required.

PDE is intermediate level calculus.

But to address the OP's question, try finding an advanced physical chemistry text. There are plenty of uses for PDEs in pchem. Your average introductory level texts won't bother to go that far into the math, probably just through you a few simple related rate equations, but when y

## Re:Standards have slipped then... (Score:5, Informative)

This might come as shocking news to you, but the typical undergraduate PDE class only scratches the surface of a rather deep and broad subject. From the examples you list, it seems that you only worked with equations for which global existence and regularity are trivial, and you have lots of conserved quantities. Many aspects of PDEs are fields of active current research, including heuristics for fluid mechanics modeling, theoretical questions concerning geometric structures on manifolds (see Yang-Mills or Seiberg-Witten equations), and integrable hierarchies. I'm not a specialist in PDEs, but I'm sure there are others who can list much more, and describe interesting open problems in detail.

Also, I should point out that the lack of a required PDE class does not necessarily mean standards have slipped. If you look at the requirements for a major in the top math departments in the US, you'll find that they have few required courses, and many options. I think these departments have decided that students should have freedom to focus on their interests after they have learned some fundamentals, and that there are other areas of mathematics, such as abstract algebra, topology, and combinatorics, that may hold their interest. I have met many mathematicians who have little experience with even the heat and wave equations, and they have done fine, because their work was not related to these questions. It is possible that the OP has taken a similar educational track.

## Re: (Score:3, Insightful)

You're exactly right - undergrad DiffEQ is more of a "Survey of Differential Equations". It's an overview of "safe" equations - most all of the work has answers that are trivial to find. My M410 professor always joked that his job was to protect us from differential equations. That being said, 300 or 400 level DiffEQ serves as a good foundation for more advanced classes in the subject.

My area of expertise is in three-dimensional electric field modeling. It's very frustrating and enlightening at the same

## Books (Score:5, Informative)

They Feynman Lectures on Physics would probably be a good place to start. It'll be basic to advanced.

http://www.amazon.com/Feynman-Lectures-Physics-including-Feynmans/dp/0805390456/ref=pd_bbs_sr_2?ie=UTF8&s=books&qid=1226900482&sr=8-2 [amazon.com]

If you want something more specific, to a topic, there will be a slew of books. I found some pretty good ones following links on Amazon from one to another and reading reviews.

## Re:Books (Score:4, Insightful)

Surely You're Jokingand I'd recommend them to anybody beginning in physics, especially to Professors of freshman physics, which is usually not so much taught as shoveled. The lectures are taken from his lessons in first year physics, so not too difficult for a math grad student with no previous physics.## Re:Books (Score:5, Informative)

I just thought of another one. It's Mathematical Methods for Physicists by Arfken. I wouldn't necessarily recommend buying it, but find one you can flip through (most university libraries have it, as do most math/physics department libraries. and I can almost guarantee that someone you know has this book).

http://www.amazon.com/Mathematical-Methods-Physicists-George-Arfken/dp/0120598760/ref=sr_1_5?ie=UTF8&s=books&qid=1226903092&sr=1-5 [amazon.com]

It's a math text, but since it's geared as a math text for physicists, the explanations may have the right amount of physics in them.

(I've always liked it as my math reference).

Though, I don't think this will be at your level (probably below), but it may help with the ground work. As I said, don't buy it, but find a copy to flip through.

## Re: (Score:3, Informative)

My academic background is in physics, so I'm likely more on the other side of the fence than you are, and still have little idea of what your expectations are book-wise.

Anyway, here's a few just to get you started that I would recommend looking into:

## Re:Books (Score:4, Informative)

My wife, who's a Math PhD who does tons of PDEs in her field (Fluid Dynamics) seconds the Charlie Harper book.

She didn't even look up from her work to answer when I hollered the question to her. She says "Matthews and Walker" too, but doesn't remember the title. I don't see it up on the shelf, so it might be in her office, which means I can't relay the exact title, but if you look for "Matthews and Walker" you will probably find it.

## Re: (Score:3, Informative)

## Re:Books (Score:4, Interesting)

As a 4th-year Physics undergrad, I have to voice my opinion that I absolutely can't stand Feynman's texts.

They're nice to glance at, but approach the subject in a considerably different manner than any of the other renowned physics texts.

Similarly, his proofs were terse to the point of being difficult to follow. I'll admit that my mathematical intuition isn't the greatest, though I can't help but think that this was intentional on Feynman's part, as to weed out those with weak mathematical skills from his freshman lectures. This makes them rather frustrating to use as a general reference. Similarly, the texts are largely theoretical, and offer little advice with regard to problem-solving.

Personally, I've had good experiences with the Landau/Lifshitz series of texts, and it's hard to go wrong with Griffith's books on EM and QM. Goldstein's text on Classical Mechanics is also a well-known classic.

That's not to say that that Feynman's texts are all bad. Some sections are outright brilliant, and he actually takes the time to explain himself rather extensively in many sections, which many physics (and math) writers frequently neglect to do. I keep a copy of all 3 volumes on my bookshelf, as they

areoccasionally handy. However, I wouldn't dream of using them as my only reference.## Re: (Score:3, Informative)

With that said I generally found that doing problems was far better at building intuition than any text. The texts out there all have their own take and generally speaking

## Re:Books (Score:4, Insightful)

I'll admit that my mathematical intuition isn't the greatest, though I can't help but think that this was intentional on Feynman's part, as to weed out those with weak mathematical skills from his freshman lectures.Bear in mind, the classes these lectures were delivered to were at Caltech in the 60s, I believe. Those with weak mathematical skills didn't get in.

Also realize that many of the undergrad lecturers at Caltech take it as a badge of honor to see how much they can shovel at the undergrads, equating density and difficulty with learning. You might find that nearly all of the students in the class were spending quite some time poring over those lectures to figure them out - not because the profs wanted to weed them out, but because that's simply how things were/are done at Caltech. On the other hand, that's something you didn't want/need to do, valuing your own time and sanity, and not staring an "F" in the face if you didn't.

I'm in the same boat, I wouldn't have stood a prayer in that environment either.

I keep a copy of all 3 volumes on my bookshelf, as they are occasionally handy. However, I wouldn't dream of using them as my only reference.Yeah, Feynmann wouldn't make a good reference but he's definitely entertaining and insightful. Probably about like Ambrose Bierce in that regard.

## Re:Books (Score:4, Insightful)

All the best, are.

## Re: (Score:2)

## He said "Mathematician" (Score:5, Insightful)

The mathematician goes off for three weeks filling in all the gaps and "leaps of faith".

He comes back to the book, and reads page three.

Mathematician flings book against the wall, and goes off and finds something more rigorous to read.

As I remember them, the Feynman lecture series were finely crafted instruments of torture for those who delight in rigor. Personally I think he entitled the wrong book "You must be Joking!"

## Re:He said "Mathematician" (Score:4, Interesting)

You've got to remember that there was an awful lot that was obvious to Feynman - hell he won the Putnam without breaking a sweat. He ran into a classmate who wondered why he wasn't taking the Putnam exam and Feynman told him he'd finished the exam. The interchange took place when there were a couple of hours left on the exam clock and none of the other contenders completed the test in the allotted time.

He felt that Mathematicians spent an awful lot of energy developing stuff that was obvious, and hence a waste of his time. He used to harangue math graduate students that if they could clearly state what they were working on, he could reproduce and finish what they were doing within the evening. The thing was, he could do it. He was far more interested in why things worked the way they did rather than proving that the math he was using was correctly applied - the results mattered to him far more than the technique.

He used to say that the renormalization techniques he used developing QED which won him the Nobel Prize probably weren't kosher math but they produced the right answer to the tenth decimal place.

In the end, that's what doomed the Feynman Physics Undergraduate books - they were simply too advanced for the vast majority of their intended audience. While he was giving the lectures, the undergraduate attendance declined while the graduate attendance increased thereby keeping the room full which misled him as to how clearly he was teaching his intended audience. It wasn't until the mid terms came in that he realized something was amiss. If the average Caltech student couldn't suss what he was saying, it's a fair bet few other physics undergrads would be able to. The graduate students, and other faculty, on the other hand, loved the class because it gave them insights into topics they thought they completely understood.

## PDEs now? (Score:5, Insightful)

You are in your third year of a PhD program and are only now studying PDEs? Aren't they more of an undergrad topic, or have schools gotten weaker? :)

p.s. First post!

## Re:PDEs now? (Score:5, Insightful)

There can be a world of difference between graduate and undergraduate PDE courses; it's not like everything that's known about PDEs can be taught in a couple of undergraduate semesters. I expect most undergrad PDE courses are geared towards showing you the methods that work for a few classes of linear PDEs; a graduate course might be concerned with the analytical underpinning of those methods, or maybe about numerical and analytic techniques that are useful in solving classes of nonlinear PDEs, etc.

That being said, though, from the way the original question is worded, it sounds like it's the first time this person has seriously encountered PDEs. Not having this happen until the third year of a PnD program does seem a little odd.

p.s. No, you're not.

## Re: (Score:2)

Unless there's some school offering a "undergrad through Ph.D" program. Then that would make sense.

## Re: (Score:2)

Ah, you're probably right; in that case, it's probably not too odd. :)

## Re: (Score:2)

I think what they meant was, "hey! hey! we're smart too!"

## Re: (Score:3, Insightful)

My question is:

How does a PhD student get that far without any physics courses?

## Re: (Score:3, Interesting)

No. ODE's are typical of Undergrad. But, PDE's are typical of Masters. That isn't to say that PDE's are taught in Undergrad, period. Rather that PDE's in Undergrad is atypical. At least in North America. Other parts of the world either have vastly superior high-school/Undergrad or skip a lot of the, necessary for actually understanding, stuff. Germany and China are respective examples.

## Re:PDEs now? (Score:5, Insightful)

PDEs are not normally part of a math degree. They do form the central basis to applied math degrees. People in the engineering and physical sciences have a great understanding of applied math, but they have little to no understanding of pure math. If you get a BS in a physical science or a BE from any decent university, you will basically have a minor in applied math (adv. calc, ODEs, PDEs, probability, statistics, nonlinear dynamics, complex analysis, and calculus of variations). But you have not even scratched the surface of pure math. Mathematicians worry primarily about pure math. To teach PDEs would be insulting to them due to its lack of generality. As many physicists and engineers have learned over time, if you have a difficulty in understanding mathematics that applies to your field, the worst person you can go to for help would be a mathematician that hasn't studied applied math. The best person you could go to would be a mathematician who specialized in applied math.

## Re: (Score:3, Informative)

Nope, I'm currently doing a second degree in my spare time and did this stuff last year (my second year) and my degree isn't even a full maths degree, only 50% of it is maths.

What on earth is a 3rd year Phd maths student doing only doing PDEs now??? This really is undergrad stuff. I understand these topics can go more advanced but the stuff described sounds like the basic undergrad stuff.

I wonder if perhaps the person asking the question actually means they're a 3rd year undergrad student who wants to do a

## Re: (Score:3, Interesting)

## Re:PDEs now? (Score:5, Informative)

You both probably studied how to solve certain simple PDEs in simple geometries (like the heat, wave, and Poisson equations). At a graduate level one normally learns how to prove existence and uniqueness of solutions to PDEs, how smooth those solutions are (i.e. how many derivatives do the solutions possess), and how to define weak forms of PDEs for which non-classical solutions exist (solutions that are not necessarily even continuous). Then there is the whole area of non-linear equations which is a very active research topic... (See the Navier-Stokes Equations.)

## Re: (Score:3, Informative)

It says right there in TFS that he hasn't studied the heat or wave equations.

## Re:PDEs now? (Score:4, Informative)

You both probably studied how to solve certain simple PDEs in simple geometries (like the heat, wave, and Poisson equations). At a graduate level one normally learns how to prove existence and uniqueness of solutions to PDEs, how smooth those solutions are (i.e. how many derivatives do the solutions possess), and how to define weak forms of PDEs for which non-classical solutions exist (solutions that are not necessarily even continuous). Then there is the whole area of non-linear equations which is a very active research topic... (See the Navier-Stokes Equations.)

Clearly graduate level approaches to PDEs differ from undergrad approaches.

However, the topics you suggested as grad level were mostly introduced to us at undergrad level (year 3 of 4 year course) in Chemical Engineering, and that was 30 years ago. Yes, we studied existence, uniqueness, and smoothness of PDE solutions. We also studied the diffusion/heat equation with moving boundaries (diffusion with reaction), and coupled instances of the diffusion equation (interphase transfer).

The Navier-Stokes equations were introduced, but not studied until graduate level. I think that generic numerical solvers are used nowadays for simple NS problems (they were PhD stuff in those days), but analytic underpinnings are reserved for grad school.

## Making the math tangible does help (Score:5, Insightful)

## Some essentials (Score:5, Informative)

Goldstein, Classical Mechanics. Standard grad level mechanics, solid book, mathematically rigorous yet still intuitive.

For EM and Quantum, even a math grad should read the advanced undergraduate books by Griffiths:

Introduction to Electrodynamics

Introduction to Quantum Mechanics

For thermodynamics, I don't know the best text.

For General Relativity, the standard undergrad book is Hartle's Gravity. But since you're a math PhD, you can go straight to the finest first grad level Relativity book by Sean Carroll:

Spacetime and Geometry

If you're looking for intuition, the indispensable and invaluable books are Feynman's Lectures on Physics.

I can recommend mathematical physics texts, but I get the impression you want the missing background for understanding. Hope this is helpful.

## Re:Some essentials (Score:5, Informative)

I'd like to second all of these recommendations, but for Quantum Mechanics if your linear algebra is sharp, I might suggest Principles of Quantum Mechanics by Shankar.

Griffifhs' Quantum Mechanics is an excellent introduction, but it assumes relatively little math knowledge, and tends to gloss over some of the assumptions being made. This is good for a student who's going to spend most of his effort trying to learn the practical aspects of doing Quantum Mechanical calculations, but not ideal for someone who grasps the math quickly and easily, and wants to really understand how things work.

Shankar is a little more difficult mathematically (and is thus often a poor introduction for an undergrad) but it very clearly lays out the assumptions being made, and how the math relates to the physics.

I haven't actually read the Sean Carroll book, but I took a course from him, and I can't imagine the book is anything but excellent.

## Re: (Score:3, Informative)

Griffiths QM book is absolutely terrible. All it does is skim the surface. Greiner is vastly superior. Griffiths E&M book is good though.

## Re: (Score:2, Insightful)

Arnold -- Mathematical Methods of Classical Mechanics. It has a very nice geometric viewpoint, and is very math oriented (published in the GTM series from Springer). No physics background required!## Re: (Score:2)

## Re: (Score:2)

The OP requests non-undergrad books for undergrads, but I wholeheartedly disagree. The graduate PDE course is covering the technical

## Re: (Score:2)

The Griffiths books are not "advanced" by any interpretation, and are exactly the sort of book that maths students would find objectionable, along with Carroll's book and Hartle's book.

For Classical mechanics, I would recommend looking at a variety of books, since most people have very strong and differing opinions on them, much like vi and emacs. I personally like Landau, but Jose and Salatan uses a more mathematically interesting perspective; I can't stand Goldstein, for some reason.

For Quantum Mechanics,

## A survey of the best (Score:3, Informative)

## My favorites (Score:3, Informative)

I think the best book for what you are asking (and I am 95% sure this is the right book, but I've lent it out so I had to look it up from dover) is "Vector Analysis" by Homer E. Nowell. It develops the theory of vector calculus using an intuitive approach and builds up the theory of electromagnetism simultaneously.

You might also look into the Feynman lectures. I do not normally recommend them as 'learning' material because, while excellent, I'm not aware that they come with any problem sets. But for you they may be a good supplement.

And, just to throw it out there, but it seems to me that most technical schools have enough overlap between physics degree requirements and math degree requirements that if you have a reasonable interest in the other it might not be out of the question to work that into your curriculum.

## Man... (Score:2)

## Re:Man... (Score:5, Funny)

No, I'm in the exact opposite situation. I don't know anything about PhD level math _or_ physics.

## Yes, stick to the mathematics. (Score:3, Insightful)

## What? (Score:4, Funny)

Why are you taking partial differential equations as a graduate student?

## Re:What? (Score:4, Funny)

## Re:What? (Score:5, Informative)

Contrary to what most people seem to think, the material taught in most Calculus and Differential Equations courses has very little resemblance to what most Mathematicians study. These fields actually all fall under the heading of Analysis, which is just one of several major branches of mathematics. A student not interested in analysis could easily spend most of his math career working in another area.

For the most part, differential equations courses are aimed at non math majors, such as physicists, chemists, engineers, and the more analytically minded biologists and economists, so even a Math major specifically interested in analysis isn't necessarily going to take classes on partial differential equations.

I myself double majored in Physics and Math, and every single course i took about differential equations was for the Physics major rather than the math Major, so I think that Math grad student could quite easily end up with a PhD without ever dealing with differential equations unless they interested him.

## Re: (Score:2)

## Re: (Score:3, Insightful)

Wow, the level of ignorance here is astounding, that you would get moderated so highly. Real PDE (as mathematicians study it) is HARD, and requires a heavy background in analysis. This is not the same as undergrad "PDE" courses.

This is like the high schooler saying "Why are you taking algebra as an undergrad" to a math major studying abstract algebra. Its the same word and the topics are related, but its not even close to the same thing.

## Some recommendations from another Math Ph.D (Score:5, Insightful)

## Re:Some recommendations from another Math Ph.D (Score:4, Interesting)

## Re: (Score:2, Informative)

Here's a good thermal book I used in my Undergrad.

http://www.amazon.com/Thermal-Physics-2nd-Charles-Kittel/dp/0716710889/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1226902024&sr=8-1 [amazon.com]

Also had a bit from http://science.slashdot.org/comments.pl?sid=1031405&op=Reply&threshold=-1&commentsort=0&mode=nested&pid=25782785 [slashdot.org]

It wasn't too bad.

Hard for me to say if either of those are really "good" texts as I hated Thermal.

## Re: (Score:2)

The OP is a graduate student in a field that isn't physics and says he never took physics anywhere. He's overestimating his abilities when he says he doesn't want to start with an undergraduate textbook because that's exactly where he should start. Unless he's cramming for an exam, he should take the time to start with college physics books and move up as he understands the material. PDE is difficult, but the basic physical concepts they represent are relatively simple to understand.

## Re:Some recommendations from another Math Ph.D (Score:4, Interesting)

Jumping Jesus on a pogo stick, you're pointing him to The Black Death straight out of the gate? Why not give him underwear made of wolverine chow? Wheeler would have died ten years ago if not for the life-giving tears of those who opened that book unprepared. That is to say, everyone.

Seriously, dial it back a bit. First, hit the Feynman lectures (stop when you get to 'partons'.) Then, for someone coming from a mathematical bent, I'd suggest starting with Sokolnikoff's book "Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua", which covers a lot of ground besides GR. Due to the absence of a just and loving god it is out of print, but surely one of the profs in a math department with a PhD program has a copy (or at minimum the library.) And there's always copies on Alibris.

And, seconding suggestions from other posters, Kittel and Kroemer's "Thermal Physics" is a good starting point on thermo, As for quantum, in the absence of all knowledge in the field I'd start with Tipler's "Modern Physics", with the goal of ramping up to Cohen-Tannoudji, Diu, and Laloe's "Quantum Mechanics".

## Physics/Astronomy Graduate student perspective (Score:3, Informative)

## Enter the Physics vs. Math Holy War. (Score:5, Funny)

I love watching this one happen.

It's funny because no matter what, the only thing a physicist and a mathematician has ever been able to agree on is magic mushrooms.

## Road to reality (Score:5, Informative)

An excellent Physics book that is very math heavy but assumes no prereqs is Penrose's Road to Reality [amazon.com]. This pretty much covers all of the main theory/formulas in cosmology, and he has 350 pages of math (much of it graduate level) to get there.

## Try this book (Score:2)

When I was still in school, we use the Quantum Mechanics from Richard W. Robinett

http://www.oup.com/us/catalog/general/subject/Physics/QuantumPhysics/?view=usa&ci=9780198530978 [oup.com]

http://www.amazon.co.uk/Quantum-Mechanics-Classical-Visualized-Examples/dp/0195092023 [amazon.co.uk]

After that would be books on solid-state

## The Feynman Lectures on Physics (Score:5, Informative)

I can not recommend these books enough. Feynman does a brilliant job of bringing the concepts of physics to life.

All together, they are quite extensive, but the individual topics are brief enough to digest in one sitting. Wether you only have a passing interest in physics, or a graduate degree in the field, you will find that there is much to appreciate in these lectures.

Even for those simply taking physics as requirement, I think that these would give you a real appreciation of the field, and probably make the classes a lot easier at that.

## Re: (Score:2)

I second that recommendation.

They are the best set of general physics books you will fond - even if they are decades old.

(Mechanics are easy? Yes, until you reach friction, which isn't really understood.)

You'll have to like math, though. :)

## They're All Targeted for Mathematicians (Score:5, Informative)

I've a couple of degrees in Physics, and I assure you, half the print in the _vast_ majority of Physics books is equations. Most physics texts seem to assume a math minor. Most Physics majors first see partial differential equations, special functions, and group theory as undergraduates. A couple of friends took partial diffeq for fun. Yeah, that's one way to know you're a nerd.

I suggest a library or a used bookstore, as these things are expensive. Here are some of the typical texts you see around on various physics topics (by author's name, because the titles are useless):

Electromagnetism:

Griffiths is a really great undergrad book, which is easy to read.

Jackson is the classic first semester grad-school book.

Math Methods of Physics:

Arfken is a classic.

Cantrell is an up and coming variant.

Thermodynamics:

Kittel is an oldie, but a goodie. Someone else prolly has a better suggestion.

General Undergrad Phenomonology:

The World Wide Web - Invented at CERN, y'know.

Halliday & Resnic is probably the easiest book to find.

Serway is newer.

Relativity:

Rindler is the standard.

Mechanics:

Goldstein is pretty easy to find.

Quantum:

Landau (yep, the same) and Lifshitz is a solid text that

hits on Shcrodinger's equation well.

Griffiths is easier to read, as is Eisberg & Resnick.

Modern Physics:

Less of an obvious choice, but it'll be a good source for more sexy topics.

A lot of partial diffeq is used in mechanics. IIRC, partial diffeq was invented to describe mechanical systems, so many of the examples are very intuitive (for you of course, not for 99.9% of the population.)

Interestingly enough, this Wikipedia link http://en.wikipedia.org/wiki/Partial_differential_equation [wikipedia.org] can take you many places, as it seems to come from the mind of a physicist more than a mathematician.

Alternately, you will probably have success finding a physics student at your relative level that has the intuitive feel, but is weak on math. You could quite a bit from each other in short order.

may the electromagnetic force be with you,

-Rick

## Re: (Score:2, Informative)

## What I want to know is... (Score:2)

Part of the

responsibilityof a University is to see that you get a broad education. If you have had no physics, you do NOT have a broad education. Period.## Re:What I want to know is... (Score:5, Funny)

If you have never taken any psychology classes, you do NOT have a broad education. Period.

If you have never taken any philosophy classes, you do NOT have a broad education. Period.

If you have never taken any accounting classes, you do NOT have a broad education. Period.

This is fun!

## Don't be an ass. Oops, sorry, too late... (Score:3, Informative)

andscience. If not, you are NOT receiving the education you need to really understand your field. Regardless of how you feel, mathematics actually relates to (and is constrained by) our physical universe. If you do not understand that, then you are not well versed in either.A degree in mathematics, from a responsible university, should include at least some physics. And of course a degree in physics requires a certain minimum of math, or

## incorrect (Score:5, Insightful)

Most areas of science strongly rely on philosophy, and most scientists understand it poorly, usually to the detriment of the technical quality of their work. You can see this all the time, from physicists publishing embarrassingly poor papers on how quantum mechanics "disproves free will" (apparently without even an undergraduate understanding of free will), to AI researchers with little background in philosophy of mind, to statisticians rediscovering the problem of induction every few years. Not to mention the very naive understanding of the "scientific method" that an intro course in philosophy of science might be useful in addressing.

In any case, pure (as opposed to applied) math has not very much to do with the hard sciences. And there is furthermore just not enough time to fit in everything people need. A good understanding of computer science is, for example, required for most technical fields these days as well, and also fairly under-taught; probably I'd put it ahead of physics in importance to most non-majors.

## Re: (Score:2)

Nonetheless, I would not say that the breadth of an education is contingent on any particular subject; an education which encompassed almost everything but that subject would undoubtedly be broad.

## Re: (Score:2)

I humbly submit that an individual with a Mathematics/English double major and a minor in Musical Theater would have a very broad education whether or not they were required to take physics.

## Re: (Score:2)

I definitely agree with the first two statements. Can you even make it through university without taking at least one class of each?

The last one's a little more specialized, or rather "applied" - it doesn't really represent a significant research area. I think the closest I came to that was a fairly broad intro level economics class.

## Re: (Score:2)

The point I'm trying to make is that no single subject can really be used as a criterion to determine whether or not an education is broad. I think the suggestion is a kind of funny.

## Re: (Score:2)

The point I'm trying to make is that no single subject can really be used as a criterion to determine whether or not an education is broad. I think the suggestion is a kind of funny.I kinda thought the idea of a broad education was that it included at least some minimal amount of a broad range of subjects, including the ones mentioned. Didn't really think of psychology, philosophy, history, and literature as an either/or kind of deal. And yes, if you haven't had a single class in one of those subjects,

## Vector Analysis (Score:2, Insightful)

## 3rd year Math PHD and only NOW learnin Partial Dif (Score:2)

## Re: (Score:2)

The difference between engineering and math is that engineering focusses on real-world problems and the bit of math required to solve them. Because there are too many other things to learn - and engineering centers on practical applications. A lot of math appears to be intellectual masturbation unless you have proper training - and lacks any trivial practical application. Until suddenly, someone might find use for it to describe something in physics. Or not. A lot of the riddles you solve as a geek are appl

## Re: (Score:2)

especially anything in the real worldSo why would this be in a math program, again? ;-)

## Spivak (Score:2)

Elementary Mechanics from a Mathematician's Viewpoint[uga.edu], which is based on his Pathway Lectures at Keio University [keio.ac.jp].## thermodynamics (Score:2)

The wave equation and diffusion equation are technically partial differential equations because of the 3 space dimensions and time, but these are simple PDEs because the three space dimensions are basically the same and the derivatives usually only appear as the Del operator, which treats each direction equally, and the boundary conditions are usually such that the constant of integration is just zero.

In thermodynamics, you actually have serious PDEs which invol

## learning by applying (Score:2)

I hope some math professors are reading this. They always seemed to think that they only needed to teach the "how", as "why" would already be obvious or would become clear. It didn't, not for me. More like that was the excuse, because actually "how" alone was much easier to teach. I studied PDEs in calculus classes, but never used them for anything. When they did come up with example uses, they were pretty contrived, and often could be solved with plain old algebra. Or they were so small that hand app

## Aerodynamics? (Score:2, Informative)

It's for senior undergraduates, but "Fundamentals of Aerodynamics" by John Anderson progresses from inviscid flow all the way through to tacking the

Navier-Stokes equations using numerical methods. I'm but a humble engineer and looking at those equations hurt my head, so it might be OK for you.

Oh, and you'll get $1M if you so happen to solve the Navier Stokes equations (or simply prove a solution exists).

## Classical Mathematical Physics (Score:2)

It is written in a mathemical language (Def, Theorem, Proof...) and is highly structured to help line out the mathematical basics behind classical mechanics and electrodynamics (some differential geometry is needed for the latter).

The second volume on quantum mechanics requires a pretty solid kn

## Book Recommendation (Score:2, Informative)

## V.I Arnold (Score:2, Interesting)

I don't know if it has been mentioned here, but V.I. Arnold (Lectures on Partial Differential Equations) might be a starting point. Arnold emphasizes physics in his writing. His introduction to classical mechanics is an absolute must for everyone interested in this kind of topics! He really blows away the fog.

## PDE is very intuitive but add a class in Q.M. (Score:2)

I think all the PDE (partial diff. eq.) like heat equation, wave equation, gravitation should have been covered in high school. Just get a basic high school text, and spend a few minutes. These equations should at any rate be so self evident that you should not have any problem to understand this intuitively.

You will probably need a textbook on quantum mechanics for the Schrodinger equation, mostly because of operator formalism or bracket notation. I would recommend just adding a class in quantum mechanics

## Lots of PDEs in this (Score:2)

Advanced Mathematical Methods for Scientists and Engineers. Carl M. Bender## Re: (Score:2)

That's the problem. Most texts that are basic physics also assume basic maths.

Maybe you can handle Jackson Electrodynamics, which is a standard graduate level text. It won't be easy, but it doesn't really assume much foreknowledge, since it lays out the groundwork in the first few chapters (which are review for

## Re: (Score:2)

I don't recommend either Halliday/Resnick/Crane or Giancoli.

I dunno, I remember finally really "getting" pdes from H&R, though maybe that was very supplemented by lectures. I

doknow that as subjects go, what really made the math click was E&M: Maxwell's Equations were just so damn elegant and beautiful it all came together there for me (though coffee cups are good for boundary value problems - I seem to remember Boyce and DePrima being a good text with enough of the physics to make it work well).## Re: (Score:2)

Giancoli isn't very good. Meh.

## Re: (Score:3, Insightful)

## Re: (Score:2)

You're right. Because *everything* that a person needs to know about PDEs is taught in that undergrad class. This must be some sort of joke! The outrage!! We shall not stand for this!

Of course, the other option (even though it's completely ridiculous) is that--like most colleges--there is more than one level of PDE class, just as there is more than one calculus class. But I know, it's crazy (that's why we threw that out at the start!)

The material he is describing is what is covered in the undergrad PDE course. Its frequently given as both an undergrad course number and a graduate course number: same book, just more work for the grad level class.

## Re: (Score:2)

## Re:3rd year PhD student taking PDE? (Score:5, Funny)

## Re:3rd year PhD student taking PDE? (Score:4, Funny)

An algebra? You mean there's more than one?

## abstract algebra for java programmers... (Score:3, Insightful)

Description about those groups and fields are like Java interfaces. These are just a collection of facts that allow you to prove theorems without knowing the particular implementation of an algebraic structure (e.g. natural numbers, matrices, geometry); or in the case of Java, being able to write a class method to use another class without looking at the actual source code of the other class.

Abstract algebra is exactly that, abstraction.

## Re: (Score:3, Interesting)

There are applications, too. The operators in quantum mechanics form a C*-algebra acting on a Hilbert space. Learning the properties of a C*-algebra is easier than trying to deduce what the prope

## Re: (Score:2)

The general idea is straightforward. Partial derivatives are just the concept of a derivative generalized to higher dimensions. Just as a derivative is a tangent to a curve, a partial derivative is is a tangent plane to a surface.

I recall studying PDEs in a 3rd year undergrad course. How you can get to Ph.D level in maths and not have at least a working (basic) understanding of them is beyond me.

## Re: (Score:3, Insightful)

I recall studying PDEs in a 3rd year undergrad course. How you can get to Ph.D level in maths and not have at least a working (basic) understanding of them is beyond me.

I'm finishing my PhD in math, and I know almost nothing about PDEs. It's not relevant to my field of research.

## Re:Partial differential equations (Score:5, Informative)

Good thing you weren't modded up. Basically nothing you said was enlightening or even correct, except for the contents of the first sentence.

You didn't even bother to correct the OP, you just sat back and decided to be a useless pedant. Yes, OP is technically incorrect, but your post is uninformative and completely worthless.

All possible partial derivatives of a point on a 3-dimensional graph fall on a tangential plane. Usually we speak of a tangent line, setting x or y constant, but if one redefines the coordinates, then any line on that plane that passes through that point is a partial derivative. So that "partial derivative plane" contains all possible partial derivatives of that point. This designation is intuitive and not particularly misleading, so there was little point in being an ass about it.